Find the number of ways in which 4 letters may be selected

Find the number of ways in which 4 letters may be selected from the word "Examination"?
A. 66
B. 70
C. 136
D. 330
E. 4264

We have, 11 letters: {A, A, E, I, I, O, M, N, N, T, X};
Out of them there are 3 pairs: {A, A}, {I, I} and {N, N}.
So, # of distinct letters is 8: {A, E, I, I, O, M, N, N, T, X}

As pointed out, there are 3 different cases possible.
{abcd} - all 4 letters are distinct: \(C^4_8=70\);
{aabc} - two letters are alike and other two are distinct: \(C^1_3*C^2_7=63\) (\(C^1_3\) is a # of ways to choose which two letters will be alike from 3 pairs and \(C^2_7\) # of ways to choose other two distinct letters from 7 letters which are left);
{aabb} - two letters are alike and other two letters are also alike: \(C^2_3=3\) (\(C^3_3\) is a # of ways of choosing two pairs of alike letter from 3 such pairs);

Total=70+63+3=136.

Answer: C.

So, as you can see you can not just pick 4 letters with \(C^4_{11}\) and then divide it by some factorial as there are 3 different cases possible and each has its own factorial correction.

Hope it's clear.
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EXAMINATION has 11 letters, and in which 'A', 'I' and 'N', all occur twice. 11C4, would have been fine if all letters were distinct.

So, we have E, X, M, T, O, (AA), (II), (NN). 8 distinct letters.

1. 4 letters selected, which are all distinct: 8C4 = 70
2. 2 letters alike, and 2 distinct (eg: AAEX) = 3C1 x 7C2 = 63
3. 2 letters alike, and 2 letters alike (eg: AAII) = 3C2 = 3

So answer is, 70 + 63 + 3 = 136.

The tricky part is getting the second one correct, ( 2 letters alike, and 2 distinct)

Cheers...

I think the problem people are having with this question is in deciphering: This is a combination question where order of 'picking' does not matter. Then why do I've to group these similar letters.

Well I had this problem too, this is how I solved it:
1. Understand what the question is asking: Is it asking about me 'picking' the letters? Or is it asking me about 'arranging' these letters differently?
Ans: Its asking about 'picking'. So there you go, 1 question down
2. DONOT forget that the formula \(nCr\) is for n 'different' things - So here, you HAVE TO group the similar letter into sets.
3. Ok, now you've grouped them, what should be done now?
Well Since, you've grouped them, you can have different cases of how u now 'pick' !
-> Now, refer to what Bunuel said in his solution, and it'll make sense

Hope this helps!!

Hi there, I got 330 and don't understand why this would be wrong.
"examination" has 11 letters out of which we are selecting 4 letters.
11C4=11!/4!7!=330. What's the source of this question?

I would be thankful if someone could point out what is wrong in my solution:
In total we have to select 4 out of 11, so its 11C4.
each of the 3 couples can be substituted within itself, so for each we have 2!=2 inner combinations.
counting out those duplicities: 11C4/(2!*2!*2!) = 41.25

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE


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Hi Bunuel,
If the ques states that no repititions are possible, we will divide by factorial of no of time the letters are repeated to get the answer?Am i correct?Kindly clarify

In this case we would be basically selecting 4 letters out of 8: {A, E, I, O, M, N, T, X}.
Answer: \(C^4_8=70\)
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When is it ok to use the anagram method for such problems?

I did 11! / 5!2!2!2! , but obviously incorrect

Any idea?

Thanks!
Cheers
J

11!/(2!2!2!) is the number of arrangements of the word "examination", which is not what the question is asking.
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@EXAMINATION: Total letters: 11 -> A {2}, I{2}, N {2}, [E, X, M, T, O {5}]

Letters to be selected are: 4

Possible cases:

Case I: All 4 letters are different: \(^8{C_4}\) = 70 [E,X,A,M,I,N,T,O]

CaseII: Two letters same and two different: \(^3{C_1}\) * \(^7{C_2}\) = 3 * 21 = 63

CaseIII: Two pairs of same letters: \(^3{C_2}\) = 3

Total ways: 70 + 63 + 3 = 136

Answer C

Hi there, hope this solution helps.